LMFDB - Embedded newform 81.4.c.b.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,4,Mod(28,81)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	81.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.77915471046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.4.c.b.28.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +O(q^{10})$	$q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +(35.0000 + 60.6218i) q^{13} +(-32.0000 + 55.4256i) q^{16} +56.0000 q^{19} +(62.5000 - 108.253i) q^{25} -160.000 q^{28} +(-154.000 - 266.736i) q^{31} +110.000 q^{37} +(260.000 - 450.333i) q^{43} +(-28.5000 - 49.3634i) q^{49} +(-280.000 + 484.974i) q^{52} +(-91.0000 + 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +1190.00 q^{73} +(224.000 + 387.979i) q^{76} +(-442.000 + 765.566i) q^{79} -1400.00 q^{91} +(665.000 - 1151.81i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4} - 20 q^{7}+O(q^{10}) $ 2 * q + 8 * q^4 - 20 * q^7	$ 2 q + 8 q^{4} - 20 q^{7} + 70 q^{13} - 64 q^{16} + 112 q^{19} + 125 q^{25} - 320 q^{28} - 308 q^{31} + 220 q^{37} + 520 q^{43} - 57 q^{49} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} + 2380 q^{73} + 448 q^{76} - 884 q^{79} - 2800 q^{91} + 1330 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 - 20 * q^7 + 70 * q^13 - 64 * q^16 + 112 * q^19 + 125 * q^25 - 320 * q^28 - 308 * q^31 + 220 * q^37 + 520 * q^43 - 57 * q^49 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 + 2380 * q^73 + 448 * q^76 - 884 * q^79 - 2800 * q^91 + 1330 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	4.00000	+	6.92820i	0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−10.0000	+	17.3205i	−0.539949	+	0.935220i	0.458957	+	0.888459i	$0.348223\pi$
$7$	−10.0000	+	17.3205i	−0.539949	+	0.935220i	−0.998906	+	0.0467610i	$0.985110\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$	35.0000	+	60.6218i	0.746712	+	1.29334i	0.949391	+	0.314098i	$0.101702\pi$
$13$	35.0000	+	60.6218i	0.746712	+	1.29334i	−0.202679	+	0.979245i	$0.564965\pi$
$14$		0			0
$15$		0			0
$16$	−32.0000	+	55.4256i	−0.500000	+	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	56.0000			0.676173			0.338086	−	0.941115i	$-0.390220\pi$
$19$	56.0000			0.676173			0.338086	+	0.941115i	$0.390220\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	62.5000	−	108.253i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	−160.000			−1.07990
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		0			0
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.837192	−	0.546908i	$-0.815805\pi$
$31$	−154.000	−	266.736i	−0.892233	−	1.54539i	−0.0550403	−	0.998484i	$-0.517529\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	110.000			0.488754			0.244377	−	0.969680i	$-0.421417\pi$
$37$	110.000			0.488754			0.244377	+	0.969680i	$0.421417\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	260.000	−	450.333i	0.922084	−	1.59710i	0.125900	−	0.992043i	$-0.459818\pi$
$43$	260.000	−	450.333i	0.922084	−	1.59710i	0.796184	−	0.605054i	$-0.206849\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−28.5000	−	49.3634i	−0.0830904	−	0.143917i
$50$		0			0
$51$		0			0
$52$	−280.000	+	484.974i	−0.746712	+	1.29334i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	−0.945584	−	0.325379i	$-0.894508\pi$
$61$	−91.0000	+	157.617i	−0.191006	+	0.330832i	0.754578	+	0.656210i	$0.227842\pi$
$62$		0			0
$63$		0			0
$64$	−512.000			−1.00000
$65$		0			0
$66$		0			0
$67$	440.000	+	762.102i	0.802307	+	1.38964i	0.918094	+	0.396362i	$0.129728\pi$
$67$	440.000	+	762.102i	0.802307	+	1.38964i	−0.115787	+	0.993274i	$0.536939\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	1190.00			1.90793			0.953966	−	0.299916i	$-0.0969588\pi$
$73$	1190.00			1.90793			0.953966	+	0.299916i	$0.0969588\pi$
$74$		0			0
$75$		0			0
$76$	224.000	+	387.979i	0.338086	+	0.585583i
$77$		0			0
$78$		0			0
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	0.358177	+	0.933654i	$0.383399\pi$
$79$	−442.000	+	765.566i	−0.629480	+	1.09029i	−0.987656	+	0.156637i	$0.949935\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$83$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−1400.00			−1.61275
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	665.000	−	1151.81i	0.696088	−	1.20566i	−0.273725	−	0.961808i	$-0.588256\pi$
$97$	665.000	−	1151.81i	0.696088	−	1.20566i	0.969813	−	0.243851i	$-0.0784109\pi$
$98$		0			0
$99$		0			0
$100$	1000.00			1.00000
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−910.000	−	1576.17i	−0.870534	−	1.50781i	−0.861446	−	0.507850i	$-0.830440\pi$
$103$	−910.000	−	1576.17i	−0.870534	−	1.50781i	−0.00908799	−	0.999959i	$-0.502893\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	−646.000			−0.567666			−0.283833	−	0.958874i	$-0.591606\pi$
$109$	−646.000			−0.567666			−0.283833	+	0.958874i	$0.591606\pi$
$110$		0			0
$111$		0			0
$112$	−640.000	−	1108.51i	−0.539949	−	0.935220i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	665.500	+	1152.68i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	1232.00	−	2133.89i	0.892233	−	1.54539i
$125$		0			0
$126$		0			0
$127$	380.000			0.265508			0.132754	−	0.991149i	$-0.457618\pi$
$127$	380.000			0.265508			0.132754	+	0.991149i	$0.457618\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−560.000	+	969.948i	−0.365099	+	0.632370i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−1288.00	−	2230.88i	−0.785948	−	1.36130i	−0.928431	−	0.371504i	$-0.878842\pi$
$139$	−1288.00	−	2230.88i	−0.785948	−	1.36130i	0.142484	−	0.989797i	$-0.454491\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	440.000	+	762.102i	0.244377	+	0.423273i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−874.000	+	1513.81i	−0.471027	+	0.815843i	−0.999451	−	0.0331378i	$-0.989450\pi$
$151$	−874.000	+	1513.81i	−0.471027	+	0.815843i	0.528424	+	0.848981i	$0.322783\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.667699	+	0.744432i	$0.267279\pi$
$157$	1925.00	+	3334.20i	0.978546	+	1.69489i	0.310847	+	0.950460i	$0.399387\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−3400.00			−1.63379			−0.816897	−	0.576783i	$-0.804308\pi$
$163$	−3400.00			−1.63379			−0.816897	+	0.576783i	$0.804308\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−1351.50	+	2340.87i	−0.615157	+	1.06548i
$170$		0			0
$171$		0			0
$172$	4160.00			1.84417
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	1250.00	+	2165.06i	0.539949	+	0.935220i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	3458.00			1.42006			0.710031	−	0.704171i	$-0.248681\pi$
$181$	3458.00			1.42006			0.710031	+	0.704171i	$0.248681\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	575.000	+	995.929i	0.214453	+	0.371443i	0.953103	−	0.302646i	$-0.0978698\pi$
$193$	575.000	+	995.929i	0.214453	+	0.371443i	−0.738650	+	0.674089i	$0.764536\pi$
$194$		0			0
$195$		0			0
$196$	228.000	−	394.908i	0.0830904	−	0.143917i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	−5236.00			−1.86518			−0.932588	−	0.360942i	$-0.882455\pi$
$199$	−5236.00			−1.86518			−0.932588	+	0.360942i	$0.882455\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4480.00			−1.49342
$209$		0			0
$210$		0			0
$211$	−3016.00	−	5223.87i	−0.984028	−	1.70439i	−0.646177	−	0.763188i	$-0.723633\pi$
$211$	−3016.00	−	5223.87i	−0.984028	−	1.70439i	−0.337852	−	0.941199i	$-0.609700\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6160.00			1.92704
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1610.00	−	2788.60i	0.483469	−	0.837393i	−0.516351	−	0.856377i	$-0.672710\pi$
$223$	1610.00	−	2788.60i	0.483469	−	0.837393i	0.999820	+	0.0189844i	$0.00604328\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	−2233.00	−	3867.67i	−0.644370	−	1.11608i	−0.984447	−	0.175684i	$-0.943786\pi$
$229$	−2233.00	−	3867.67i	−0.644370	−	1.11608i	0.340076	−	0.940398i	$-0.389547\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	3689.00	−	6389.54i	0.986014	−	1.70783i	0.348673	−	0.937244i	$-0.386632\pi$
$241$	3689.00	−	6389.54i	0.986014	−	1.70783i	0.637341	−	0.770582i	$-0.280034\pi$
$242$		0			0
$243$		0			0
$244$	−1456.00			−0.382012
$245$		0			0
$246$		0			0
$247$	1960.00	+	3394.82i	0.504906	+	0.874523i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2048.00	−	3547.24i	−0.500000	−	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−1100.00	+	1905.26i	−0.263902	+	0.457092i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3520.00	+	6096.82i	−0.802307	+	1.38964i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	812.000			0.182013			0.0910064	−	0.995850i	$-0.470992\pi$
$271$	812.000			0.182013			0.0910064	+	0.995850i	$0.470992\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2015.00	−	3490.08i	0.437074	−	0.757035i	−0.560388	−	0.828230i	$-0.689348\pi$
$277$	2015.00	−	3490.08i	0.437074	−	0.757035i	0.997462	+	0.0711951i	$0.0226813\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	−0.994476	−	0.104961i	$-0.966528\pi$
$283$	−2800.00	−	4849.74i	−0.588137	−	1.01868i	0.406340	−	0.913722i	$-0.366805\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$	4760.00	+	8244.56i	0.953966	+	1.65232i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	5200.00	+	9006.66i	0.995758	+	1.72470i
$302$		0			0
$303$		0			0
$304$	−1792.00	+	3103.84i	−0.338086	+	0.585583i
$305$		0			0
$306$		0			0
$307$	10640.0			1.97804			0.989018	−	0.147797i	$-0.0472182\pi$
$307$	10640.0			1.97804			0.989018	+	0.147797i	$0.0472182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−5005.00	+	8668.91i	−0.903832	+	1.56548i	−0.0813539	+	0.996685i	$0.525924\pi$
$313$	−5005.00	+	8668.91i	−0.903832	+	1.56548i	−0.822478	+	0.568797i	$0.807409\pi$
$314$		0			0
$315$		0			0
$316$	−7072.00			−1.25896
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	8750.00			1.49342
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−496.000	+	859.097i	−0.0823644	+	0.142659i	−0.904265	−	0.426971i	$-0.859580\pi$
$331$	−496.000	+	859.097i	−0.0823644	+	0.142659i	0.821901	+	0.569631i	$0.192914\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	2465.00	+	4269.51i	0.398448	+	0.690133i	0.993535	−	0.113529i	$-0.0362155\pi$
$337$	2465.00	+	4269.51i	0.398448	+	0.690133i	−0.595086	+	0.803662i	$0.702882\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−5720.00			−0.900440
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	5957.00	−	10317.8i	0.913670	−	1.58252i	0.104834	−	0.994490i	$-0.466569\pi$
$349$	5957.00	−	10317.8i	0.913670	−	1.58252i	0.808837	−	0.588033i	$-0.200098\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3723.00			−0.542790
$362$		0			0
$363$		0			0
$364$	−5600.00	−	9699.48i	−0.806373	−	1.39668i
$365$		0			0
$366$		0			0
$367$	−2170.00	+	3758.55i	−0.308646	+	0.534591i	−0.978066	−	0.208293i	$-0.933209\pi$
$367$	−2170.00	+	3758.55i	−0.308646	+	0.534591i	0.669420	+	0.742884i	$0.266543\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6175.00	−	10695.4i	−0.857183	−	1.48469i	−0.874605	−	0.484837i	$-0.838879\pi$
$373$	−6175.00	−	10695.4i	−0.857183	−	1.48469i	0.0174213	−	0.999848i	$-0.494454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8584.00			−1.16340			−0.581702	−	0.813402i	$-0.697613\pi$
$379$	−8584.00			−1.16340			−0.581702	+	0.813402i	$0.697613\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	10640.0			1.39218
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1190.00			0.150439			0.0752196	−	0.997167i	$-0.476034\pi$
$397$	1190.00			0.150439			0.0752196	+	0.997167i	$0.476034\pi$
$398$		0			0
$399$		0			0
$400$	4000.00	+	6928.20i	0.500000	+	0.866025i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	10780.0	−	18671.5i	1.33248	−	2.30793i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4123.00	−	7141.25i	−0.498458	−	0.863354i	0.501541	−	0.865134i	$-0.332767\pi$
$409$	−4123.00	−	7141.25i	−0.498458	−	0.863354i	−0.999998	+	0.00177990i	$0.999433\pi$
$410$		0			0
$411$		0			0
$412$	7280.00	−	12609.3i	0.870534	−	1.50781i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−8569.00	+	14841.9i	−0.991989	+	1.71818i	−0.386597	+	0.922249i	$0.626350\pi$
$421$	−8569.00	+	14841.9i	−0.991989	+	1.71818i	−0.605392	+	0.795927i	$0.706984\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1820.00	−	3152.33i	−0.206267	−	0.357265i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−2590.00			−0.287454			−0.143727	−	0.989617i	$-0.545909\pi$
$433$	−2590.00			−0.287454			−0.143727	+	0.989617i	$0.545909\pi$
$434$		0			0
$435$		0			0
$436$	−2584.00	−	4475.62i	−0.283833	−	0.491613i
$437$		0			0
$438$		0			0
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	0.100728	+	0.994914i	$0.467883\pi$
$439$	−7462.00	+	12924.6i	−0.811257	+	1.40514i	−0.911985	+	0.410224i	$0.865450\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5120.00	−	8868.10i	0.539949	−	0.935220i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	0.332513	+	0.943099i	$0.392103\pi$
$457$	−6355.00	+	11007.2i	−0.650491	+	1.12668i	−0.983004	+	0.183585i	$0.941230\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	9890.00	+	17130.0i	0.992716	+	1.71943i	0.600698	+	0.799476i	$0.294889\pi$
$463$	9890.00	+	17130.0i	0.992716	+	1.71943i	0.392017	+	0.919958i	$0.371777\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−17600.0			−1.73282
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3500.00	−	6062.18i	0.338086	−	0.585583i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	3850.00	+	6668.40i	0.364958	+	0.632126i
$482$		0			0
$483$		0			0
$484$	−5324.00	+	9221.44i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	20900.0			1.94470			0.972351	−	0.233526i	$-0.0750265\pi$
$487$	20900.0			1.94470			0.972351	+	0.233526i	$0.0750265\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	19712.0			1.78447
$497$		0			0
$498$		0			0
$499$	7568.00	+	13108.2i	0.678938	+	1.17596i	0.975301	+	0.220880i	$0.0708930\pi$
$499$	7568.00	+	13108.2i	0.678938	+	1.17596i	−0.296363	+	0.955075i	$0.595774\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1520.00	+	2632.72i	0.132754	+	0.229937i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−11900.0	+	20611.4i	−1.03019	+	1.78433i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−12040.0			−1.00664			−0.503320	−	0.864100i	$-0.667888\pi$
$523$	−12040.0			−1.00664			−0.503320	+	0.864100i	$0.667888\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	6083.50	−	10536.9i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−8960.00			−0.730198
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−22678.0			−1.80222			−0.901112	−	0.433586i	$-0.857248\pi$
$541$	−22678.0			−1.80222			−0.901112	+	0.433586i	$0.857248\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−820.000	+	1420.28i	−0.0640963	+	0.111018i	−0.896293	−	0.443463i	$-0.853750\pi$
$547$	−820.000	+	1420.28i	−0.0640963	+	0.111018i	0.832196	+	0.554481i	$0.187083\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8840.00	−	15311.3i	−0.679774	−	1.17740i
$554$		0			0
$555$		0			0
$556$	10304.0	−	17847.1i	0.785948	−	1.36130i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$	36400.0			2.75413
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	−0.877320	−	0.479905i	$-0.840671\pi$
$571$	−11656.0	−	20188.8i	−0.854270	−	1.47964i	0.0230498	−	0.999734i	$-0.492662\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−17710.0			−1.27778			−0.638888	−	0.769300i	$-0.720605\pi$
$577$	−17710.0			−1.27778			−0.638888	+	0.769300i	$0.720605\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$	−8624.00	−	14937.2i	−0.603303	−	1.04495i
$590$		0			0
$591$		0			0
$592$	−3520.00	+	6096.82i	−0.244377	+	0.423273i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	14651.0	−	25376.3i	0.994387	−	1.72233i	0.405567	−	0.914065i	$-0.367074\pi$
$601$	14651.0	−	25376.3i	0.994387	−	1.72233i	0.588820	−	0.808264i	$-0.299593\pi$
$602$		0			0
$603$		0			0
$604$	−13984.0			−0.942054
$605$		0			0
$606$		0			0
$607$	14210.0	+	24612.4i	0.950191	+	1.64578i	0.745007	+	0.667056i	$0.232446\pi$
$607$	14210.0	+	24612.4i	0.950191	+	1.64578i	0.205184	+	0.978723i	$0.434221\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	17390.0			1.14580			0.572900	−	0.819625i	$-0.305818\pi$
$613$	17390.0			1.14580			0.572900	+	0.819625i	$0.305818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$	13328.0	−	23084.8i	0.865424	−	1.49896i	−0.00120126	−	0.999999i	$-0.500382\pi$
$619$	13328.0	−	23084.8i	0.865424	−	1.49896i	0.866625	−	0.498959i	$-0.166284\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7812.50	−	13531.6i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−15400.0	+	26673.6i	−0.978546	+	1.69489i
$629$		0			0
$630$		0			0
$631$	1892.00			0.119365			0.0596825	−	0.998217i	$-0.480991\pi$
$631$	1892.00			0.119365			0.0596825	+	0.998217i	$0.480991\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1995.00	−	3455.44i	0.124089	−	0.214929i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−6580.00	−	11396.9i	−0.403561	−	0.698989i	0.590592	−	0.806971i	$-0.298894\pi$
$643$	−6580.00	−	11396.9i	−0.403561	−	0.698989i	−0.994153	+	0.107982i	$0.965561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−13600.0	−	23555.9i	−0.816897	−	1.41491i
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$659$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$660$		0			0
$661$	10241.0	+	17737.9i	0.602615	+	1.04376i	0.992423	+	0.122864i	$0.0392080\pi$
$661$	10241.0	+	17737.9i	0.602615	+	1.04376i	−0.389808	+	0.920896i	$0.627459\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−12025.0	+	20827.9i	−0.688751	+	1.19295i	0.283491	+	0.958975i	$0.408508\pi$
$673$	−12025.0	+	20827.9i	−0.688751	+	1.19295i	−0.972242	+	0.233977i	$0.924826\pi$
$674$		0			0
$675$		0			0
$676$	−21624.0			−1.23031
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	13300.0	+	23036.3i	0.751704	+	1.30199i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	16640.0	+	28821.3i	0.922084	+	1.59710i
$689$		0			0
$690$		0			0
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	−0.555460	−	0.831543i	$-0.687458\pi$
$691$	8036.00	−	13918.8i	0.442408	−	0.766273i	0.997868	+	0.0652705i	$0.0207910\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−10000.0	+	17320.5i	−0.539949	+	0.935220i
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	6160.00			0.330482
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−18073.0	+	31303.4i	−0.957328	+	1.65814i	−0.228381	+	0.973572i	$0.573343\pi$
$709$	−18073.0	+	31303.4i	−0.957328	+	1.65814i	−0.728948	+	0.684569i	$0.759990\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	36400.0			1.88018
$722$		0			0
$723$		0			0
$724$	13832.0	+	23957.7i	0.710031	+	1.22981i
$725$		0			0
$726$		0			0
$727$	5390.00	−	9335.75i	0.274971	−	0.476264i	−0.695157	−	0.718858i	$-0.744665\pi$
$727$	5390.00	−	9335.75i	0.274971	−	0.476264i	0.970128	+	0.242594i	$0.0779984\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−7525.00	−	13033.7i	−0.379184	−	0.656767i	0.611759	−	0.791044i	$-0.290462\pi$
$733$	−7525.00	−	13033.7i	−0.379184	−	0.656767i	−0.990944	+	0.134277i	$0.957129\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	31376.0			1.56182			0.780910	−	0.624644i	$-0.214756\pi$
$739$	31376.0			1.56182			0.780910	+	0.624644i	$0.214756\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	11726.0	+	20310.0i	0.569757	+	0.986849i	0.996590	+	0.0825179i	$0.0262962\pi$
$751$	11726.0	+	20310.0i	0.569757	+	0.986849i	−0.426832	+	0.904331i	$0.640370\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−41470.0			−1.99109			−0.995543	−	0.0943039i	$-0.969937\pi$
$757$	−41470.0			−1.99109			−0.995543	+	0.0943039i	$0.969937\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	6460.00	−	11189.0i	0.306511	−	0.530892i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2303.00	+	3988.91i	0.107995	+	0.187053i	0.914958	−	0.403549i	$-0.132224\pi$
$769$	2303.00	+	3988.91i	0.107995	+	0.187053i	−0.806963	+	0.590602i	$0.798890\pi$
$770$		0			0
$771$		0			0
$772$	−4600.00	+	7967.43i	−0.214453	+	0.371443i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	−38500.0			−1.78447
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	3648.00			0.166181
$785$		0			0
$786$		0			0
$787$	−21700.0	−	37585.5i	−0.982874	−	1.70239i	−0.651029	−	0.759053i	$-0.725662\pi$
$787$	−21700.0	−	37585.5i	−0.982874	−	1.70239i	−0.331844	−	0.943334i	$-0.607671\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12740.0			−0.570505
$794$		0			0
$795$		0			0
$796$	−20944.0	−	36276.1i	−0.932588	−	1.61529i
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	39368.0			1.70456			0.852280	−	0.523087i	$-0.175220\pi$
$811$	39368.0			1.70456			0.852280	+	0.523087i	$0.175220\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	14560.0	−	25218.7i	0.623488	−	1.07991i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	6110.00	+	10582.8i	0.258786	+	0.448231i	0.965917	−	0.258852i	$-0.0833441\pi$
$823$	6110.00	+	10582.8i	0.258786	+	0.448231i	−0.707131	+	0.707083i	$0.750011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	17066.0			0.714990			0.357495	−	0.933915i	$-0.383631\pi$
$829$	17066.0			0.714990			0.357495	+	0.933915i	$0.383631\pi$
$830$		0			0
$831$		0			0
$832$	−17920.0	−	31038.4i	−0.746712	−	1.29334i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	12194.5	+	21121.5i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	24128.0	−	41790.9i	0.984028	−	1.70439i
$845$		0			0
$846$		0			0
$847$	−26620.0			−1.07990
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	23345.0	−	40434.7i	0.937066	−	1.62305i	0.166159	−	0.986099i	$-0.446863\pi$
$853$	23345.0	−	40434.7i	0.937066	−	1.62305i	0.770907	−	0.636948i	$-0.219803\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−15652.0	−	27110.1i	−0.621699	−	1.07681i	−0.989169	−	0.146779i	$-0.953109\pi$
$859$	−15652.0	−	27110.1i	−0.621699	−	1.07681i	0.367470	−	0.930035i	$-0.380224\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	24640.0	+	42677.7i	0.963521	+	1.66887i
$869$		0			0
$870$		0			0
$871$	−30800.0	+	53347.2i	−1.19818	+	2.07532i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.708330	−	0.705881i	$-0.750551\pi$
$877$	−25075.0	−	43431.2i	−0.965476	−	1.67225i	−0.257146	−	0.966373i	$-0.582782\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−20680.0			−0.788151			−0.394076	−	0.919078i	$-0.628935\pi$
$883$	−20680.0			−0.788151			−0.394076	+	0.919078i	$0.628935\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−3800.00	+	6581.79i	−0.143361	+	0.248308i
$890$		0			0
$891$		0			0
$892$	25760.0			0.966938
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−22420.0	+	38832.6i	−0.820776	+	1.42163i	0.0843291	+	0.996438i	$0.473125\pi$
$907$	−22420.0	+	38832.6i	−0.820776	+	1.42163i	−0.905105	+	0.425188i	$0.860208\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	17864.0	−	30941.4i	0.644370	−	1.11608i
$917$		0			0
$918$		0			0
$919$	2756.00			0.0989250			0.0494625	−	0.998776i	$-0.484249\pi$
$919$	2756.00			0.0989250			0.0494625	+	0.998776i	$0.484249\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	6875.00	−	11907.8i	0.244377	−	0.423273i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−1596.00	−	2764.35i	−0.0561834	−	0.0973126i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−55510.0			−1.93536			−0.967680	−	0.252181i	$-0.918852\pi$
$937$	−55510.0			−1.93536			−0.967680	+	0.252181i	$0.918852\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$	41650.0	+	72139.9i	1.42467	+	2.46761i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−32536.5	+	56354.9i	−1.09216	+	1.89167i
$962$		0			0
$963$		0			0
$964$	59024.0			1.97203
$965$		0			0
$966$		0			0
$967$	25010.0	+	43318.6i	0.831714	+	1.44057i	0.896678	+	0.442683i	$0.145973\pi$
$967$	25010.0	+	43318.6i	0.831714	+	1.44057i	−0.0649641	+	0.997888i	$0.520693\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	51520.0			1.69749
$974$		0			0
$975$		0			0
$976$	−5824.00	−	10087.5i	−0.191006	−	0.330832i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−15680.0	+	27158.6i	−0.504906	+	0.874523i
$989$		0			0
$990$		0			0
$991$	−45628.0			−1.46258			−0.731292	−	0.682064i	$-0.761082\pi$
$991$	−45628.0			−1.46258			−0.731292	+	0.682064i	$0.761082\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−14455.0	+	25036.8i	−0.459172	+	0.795309i	−0.998917	−	0.0465191i	$-0.985187\pi$
$997$	−14455.0	+	25036.8i	−0.459172	+	0.795309i	0.539745	+	0.841828i	$0.318520\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.4.c.b.55.1		2
3.2	odd	2	CM	81.4.c.b.55.1		2
9.2	odd	6		9.4.a.a.1.1	✓	1
9.4	even	3	inner	81.4.c.b.28.1		2
9.5	odd	6	inner	81.4.c.b.28.1		2
9.7	even	3		9.4.a.a.1.1	✓	1
36.7	odd	6		144.4.a.d.1.1		1
36.11	even	6		144.4.a.d.1.1		1
45.2	even	12		225.4.b.g.199.2		2
45.7	odd	12		225.4.b.g.199.2		2
45.29	odd	6		225.4.a.d.1.1		1
45.34	even	6		225.4.a.d.1.1		1
45.38	even	12		225.4.b.g.199.1		2
45.43	odd	12		225.4.b.g.199.1		2
63.2	odd	6		441.4.e.i.361.1		2
63.11	odd	6		441.4.e.i.226.1		2
63.16	even	3		441.4.e.i.361.1		2
63.20	even	6		441.4.a.f.1.1		1
63.25	even	3		441.4.e.i.226.1		2
63.34	odd	6		441.4.a.f.1.1		1
63.38	even	6		441.4.e.j.226.1		2
63.47	even	6		441.4.e.j.361.1		2
63.52	odd	6		441.4.e.j.226.1		2
63.61	odd	6		441.4.e.j.361.1		2
72.11	even	6		576.4.a.l.1.1		1
72.29	odd	6		576.4.a.m.1.1		1
72.43	odd	6		576.4.a.l.1.1		1
72.61	even	6		576.4.a.m.1.1		1
99.43	odd	6		1089.4.a.g.1.1		1
99.65	even	6		1089.4.a.g.1.1		1
117.25	even	6		1521.4.a.g.1.1		1
117.38	odd	6		1521.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.4.a.a.1.1	✓	1	9.2	odd	6
9.4.a.a.1.1	✓	1	9.7	even	3
81.4.c.b.28.1		2	9.4	even	3	inner
81.4.c.b.28.1		2	9.5	odd	6	inner
81.4.c.b.55.1		2	1.1	even	1	trivial
81.4.c.b.55.1		2	3.2	odd	2	CM
144.4.a.d.1.1		1	36.7	odd	6
144.4.a.d.1.1		1	36.11	even	6
225.4.a.d.1.1		1	45.29	odd	6
225.4.a.d.1.1		1	45.34	even	6
225.4.b.g.199.1		2	45.38	even	12
225.4.b.g.199.1		2	45.43	odd	12
225.4.b.g.199.2		2	45.2	even	12
225.4.b.g.199.2		2	45.7	odd	12
441.4.a.f.1.1		1	63.20	even	6
441.4.a.f.1.1		1	63.34	odd	6
441.4.e.i.226.1		2	63.11	odd	6
441.4.e.i.226.1		2	63.25	even	3
441.4.e.i.361.1		2	63.2	odd	6
441.4.e.i.361.1		2	63.16	even	3
441.4.e.j.226.1		2	63.38	even	6
441.4.e.j.226.1		2	63.52	odd	6
441.4.e.j.361.1		2	63.47	even	6
441.4.e.j.361.1		2	63.61	odd	6
576.4.a.l.1.1		1	72.11	even	6
576.4.a.l.1.1		1	72.43	odd	6
576.4.a.m.1.1		1	72.29	odd	6
576.4.a.m.1.1		1	72.61	even	6
1089.4.a.g.1.1		1	99.43	odd	6
1089.4.a.g.1.1		1	99.65	even	6
1521.4.a.g.1.1		1	117.25	even	6
1521.4.a.g.1.1		1	117.38	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +O(q^{10})\)	\(q+(4.00000 + 6.92820i) q^{4} +(-10.0000 + 17.3205i) q^{7} +(35.0000 + 60.6218i) q^{13} +(-32.0000 + 55.4256i) q^{16} +56.0000 q^{19} +(62.5000 - 108.253i) q^{25} -160.000 q^{28} +(-154.000 - 266.736i) q^{31} +110.000 q^{37} +(260.000 - 450.333i) q^{43} +(-28.5000 - 49.3634i) q^{49} +(-280.000 + 484.974i) q^{52} +(-91.0000 + 157.617i) q^{61} -512.000 q^{64} +(440.000 + 762.102i) q^{67} +1190.00 q^{73} +(224.000 + 387.979i) q^{76} +(-442.000 + 765.566i) q^{79} -1400.00 q^{91} +(665.000 - 1151.81i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4} - 20 q^{7}+O(q^{10}) \) 2 * q + 8 * q^4 - 20 * q^7	\( 2 q + 8 q^{4} - 20 q^{7} + 70 q^{13} - 64 q^{16} + 112 q^{19} + 125 q^{25} - 320 q^{28} - 308 q^{31} + 220 q^{37} + 520 q^{43} - 57 q^{49} - 560 q^{52} - 182 q^{61} - 1024 q^{64} + 880 q^{67} + 2380 q^{73} + 448 q^{76} - 884 q^{79} - 2800 q^{91} + 1330 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 - 20 * q^7 + 70 * q^13 - 64 * q^16 + 112 * q^19 + 125 * q^25 - 320 * q^28 - 308 * q^31 + 220 * q^37 + 520 * q^43 - 57 * q^49 - 560 * q^52 - 182 * q^61 - 1024 * q^64 + 880 * q^67 + 2380 * q^73 + 448 * q^76 - 884 * q^79 - 2800 * q^91 + 1330 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more